L '.'• 3. MAk • .-i 7 IC/68/104 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS FORMULATION OP QUANTUM DYNAMICS IN TERMS OF GENERALIZED SYMMETRIES CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY IN TERMS OF GROUP REPRESENTATIONS A.O. BARUT 196 9 MIRAMARE - TRIESTE
64
Embed
Barut a.O. - Formulation of Quantum Dynamics in Terms of Generalized Symmetries
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
L '.'•
3. MAk• .-i 7
IC/68/104
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICALPHYSICS
FORMULATION OP QUANTUM DYNAMICS
IN TERMS OF GENERALIZED SYMMETRIES
CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY
IN TERMS OF GROUP REPRESENTATIONS
A.O. BARUT
196 9
MIRAMARE - TRIESTE
IC/68/104
(Limited distribution)
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
FORMULATION OF QUANTUM DYNAMICS
IN TERMS OF GENERALIZED SYMMETRIES *
CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY
IN TERMS OF GROUP REPRESENTATIONS
A.O. BARUT**
MIRAMARE - TRIESTE
December 1968
* Lectures given at the Advanced School of Physics, Trieste, 1968-1969.
* * On leave of absence from the University of Colorado, Boulder, Colo. , USA.
page
APPENDIX I
WIGNER'S THEOREM 25
APPENDIX II
COVERING GROUP, PROJECTIVE AND RAYREPRESENTATIONS, GROUP EXTENSION 27
APPENDIX III
WHEN UNITARY AND WHEN ANTI-UNITARYOPERATORS? 30
APPENDIX IV
THE GROUPS O(3), SO(3), SU(2), THEIRREPRESENTATIONS AND THE QUANTUM THEORYOF ANGULAR MOMENTUM 31
- ii -
FORMULATION OF QUANTUM DYNAMICS
IN TERMS OF GENERALIZED SYMMETRIES
CHAPTER I - THE FRAMEWORK OF QUANTUM THEORY
IN TERMS OF GROUP REPRESENTATIONS
INTRODUCTION
The theory of group representations, until very recently,
has been used in quantum theory to obtain essentially kinematical
relationships. Quantum dynamics, that is, the theory of the
interactions of quantum systems, was not in any way related to
symmetry considerations, but specific "interactions" had to be
formulated in Hamiltonian or Lagrangian formalism. The
formulation of dynamics based on the concept of "system 1 + system 2
+ interaction" works satisfactorily in non-relativistic theories, but
much less satisfactorily in relativistic theories. Experience hasus
shown/that in the latter case the theories always lead to procedures
involving infinitely many steps with unknown convergence properties,
as in perturbation theory or in S-matrix theory. One can then ask
if the concept of "interaction" could not advantageously be abandoned
(even if it can be shown to exist on the basis of general principles)
and the interacting systems be treated as a whole (e.g., a "dressed"
particle, or a composite system) globally in terms of their total
quantum numbers. This idea might seem at first to be less fundamental
or exact, for we are used to considering a theory fundamental if it is
formulated in terms of few basic fields or particles and of their
interactions. One can show on the basis of special examples (e. g.
positronium) that this is not so. The global point of view has some
definite advantages not possessed by the so-called microscopic point
of view.
- iii
It is in such global theories in terms of the total quantum
numbers of the interacting systems that the concept of generalized
symmetries and the theory of group representations re-enter into
the quantum theory, this time in the formulation of dynamics. The
subject of this essay is to try to give a physical interpretation of the
dynamical groups and the use of their representations in the
formulation of quantum dynamics.
The first part of Chapter I is a review of the general
principles of quantum theory necessary for the second part, where
the notion of the symmetries of system plus interaction is developed.
The further chapters will deal in detail with specific applications.
Since von Neumann's "Mathematical Foundations of Quantum
Theory", it is customary to start with a very general framework
of quantum theory: the Hilbert space of states, all Hermitian
operators as observables, etc. One then specializes to simple systems.
In these notes we follow the opposite direction. The emphasis is
on the determination of a concrete Hilbert space, in which scalar
products can be calculated explicitly and a number obtained. We
then start from the concrete Hilbert spaces of simple systems and
enlarge them systematically to define more complex systems, always
working with specific bases of states and with the specific form of
the observables. We have further adopted an S-matrix point of
view for the observable physical processes and are not treating
the time evolution of the system.
- iv -
1. KINEMATICAL POSTULATES
1.1 States and rays
The basic framework of the quantum theoretical description of
a physical system is a Hilbert space *% whose unit rays are in one-
to-one correspondence with the "pure" states of the system. A unit
ray ¥ is the set of vectors {Mi}, [| |j = 1, X = e1", ipe $ . The
reason for the introduction of rays rather than vectors themselves
lies a) in the use of a space over the complex numbers, and b) in
the basic probability interpretation of quantum theory. The quantities
related to observable effects are the absolute values of the scalar
products, |(^(^)| ,
characterizing a ray.
products, |(^(^)| , which are independent of the parameters X, X'
1. 2 Superposition principle
This basic correspondence incorporates the superposition
principle of quantum theory. For although both in classical and
quantum theory we have to do with a continuously infinite set of
states, in quantum theory there is a set of basis states out of which
arbitrary states can be constructed by linear superposition. Thus
ti1 = ) a 0 is another vector, so that the ray { X^1 } correspondsa £_, n n ; <x
n •to another possible state of the systems, if the rays \\ip t , n = 1, 2 . . .describe physical s tates . Note that 0* - \ a 0 and X01 represent
Z ot / , n n cc
a (X 0 ) is a H different state, in general,
although 0 and (X 0 ) represent the same state. This is the famous
problem of relative phases. Such states as can be obtained from each
other by linear superposition are called "coherent" states. [ G. C. Wick,
A.S. Wightman and E. P. Wigner, Phys. Rev. 88» 101(1952)].
- 1 -
!• 3 Supers election rules
In general, there are limitations on the superposition principle.
One cannot form pure states out of the superposition of certain states;
for example, one cannot form a pure state consisting of a positively
and a negatively charged particle, or a pure state consisting of a
fermion and a boson. This does not mean that two such states cannot
interact t it only means that their formal linear combination is not
a realizable pure state. The existence of superselection rules is
connected with the measurability of the relative phase of such a
superposition and depends on further properties of the system, like
charge, baryon number, etc. The superselection rule on fermions
(i. e., separation of states of integral and half integral fermions)
follows from the rotational invariance [ see Sec. 1.13 ] . In such a
case one divides the Hilbert space % into subsets, such that the
superposition principle holds within each subset. These subsets are
called "coherent subspaces". In each subspace, ) a i) and
Z y L> n n ^a th correspond to the same state, but ) a ih and ) a1 \b ,
n*n * • /_, n^n Z_J n n
in general, correspond to distinct states.
1. 4 Probability interpretation
The physical experiments consist in preparing definite states,
in letting them interact and in observing the rate of occurrence of
other well-defined states. The transition probability between two
states is defined by |(^, if>)\ (we can also say the transition
probability between two rays ¥ and $ because this quantity is the
same for all vectors of the rays). If 0and $ are themselves linear
combinations of some basis vectors, then the phase problem is inside
these quantities. This quantity can be related, by multiplying it with
certain kinematical factors, to the observed quantities like cross-*)
sections and lifetimes .
^ For general cross-section formulas see: G. Kallen, Elementary Particle
Physics (Addison-Wesley, 1964), Ch. I;
A.O. Barut. The Theory of the Scattering Matrix, (Macmillan 1967), Ch.III.
- 2 -
1. 5 The dynamical problem
Now in order to evaluate quantities like | {ip, 0) | we must
have a definite realization of the Hilbert space %• , and must obtain a
number that can be compared with experiment, i. e., a definite
labelling of the states ip, <!> . . . and a definite expression for the
scalar product. We shall refer to this realization as the concrete
Hilbert space. This is the more important and the more difficult
part of the theory. Although all Hilbert spaces of the same dimension
are isomorphic and one can transform one realization into another,
some definite explicit realization with a physical correspondence is
necessary.
If the Hilbert space framework is the kinematical principle of
quantum theory, the explicit calculation of states ^, 0, , » , , or, of
scalar products (ijj, <f>),is the dynamical part of quantum theory.
In simple cases, the dynamical problem is solved by postulating
an equation for the states \p,<f>. . . and identifying all solutions of
the equation with all the states of the physical system, such as in the
Schrbdinger theory. For more complicated systems, or for unknown
new systems, this is not possible. Even if we know all the states of
an isolated system, measurements on the system are carried out by
additional external interactions which change the system.
Short of the complete calculation of scalar products (^,i>), some
very general principles allow one to derive a number of important
properties of these quantities. It is along these lines that the
traditional use of group representations in quantum theory has been
developed *', More recently the quantum theoretical Hilbert space
has been identified with the explicit carrier space of the representations
of general groups and algebras. In this second sense the group
representations solve the dynamical problem. We shall explain
both of these aspects.
*) E. P. Wigner, Group Theory. (Academic Press 1959).
- 3 -
1. 6 Equivalent descriptions
First of all, as in any correspondence, let us deal with the
equivalent mappings between the physical states and the rays in
Hilbert space. For the knowledge of physically equivalent descriptions
of a system reflects already, as we shall see, important properties of
the system.
If the same physical system can be described in two different
ways in the same coherent subspace of the Hilbert space, once by
rays ¥..,0, . . . , and once by rays Y , $ , . . . (for example, by two
different observers), the same physical state once described by Y-,>
in the other case by Y , then the transition probabilities must be the
same by definition of equivalence. What is then the correspondence
between the raysH and TF ? Mathematically it is more convenient to
find out the corresponding map between the vectors \p, <p, . . . in the
Hilbert space. Because only the absolute values are invariant, the
transformation in the Hilbert space can be unitary or anti-unitary.
In fact, one can prove rigorously (see Appendix I, Wigner's theorem)
that one can choose vectors 0,,<p,, . . . from rays ¥ . $ * . . . in the
first description and vectors ^9,q> , . . . from the rays ¥ , O , . . .
in the second description,such that the correspondence between
ip , . . , and qfr # t . t is either unitary or a nti-unitary. That is, one
can construct a unitary or anti-unitary correspondence. These two
possibilities come from the fact that the complex field has two {and
only two) automorphisms that preserve the absolute values: the
identity automorphism and the complex conjugation. For the Hilbert
space over a real field Wigner's theorem yields only unitary
transformations (up to a phase), because the only automorphism of the
real field is the identity automorphism. In fact, Wigner's theorem is
closely related to the fundamental theorem of protective geometry.
[J. S. Lomont and P. Mendelson, Ann. Math. _78, 548 (1963)] .
One or the other case occurs for a given situation. Whether the
transformation is unitary or anti-unitary depends on further properties
of the two equivalent descriptions of the system. It does not depend,
however, on the choice of vectors ^, <p, . . . from the rays: if the
- 4 -
transformation is, for example, unitary for a choice ,,<p , . . .
there is no other choice Xip , A'cp , . . . such that it becomes anti-
unitary and vice_versa_. [ E. P. Wigner, J. Math. Phys. _1, 409, 414
(I960)]. Furthermore, once a vector tyn is chosen, the others,
<po, X , . . . are uniquely determined from the requirement that the
correspondence is unitary (or anti-unitary).
1. 7 Symmetry transformations
The description of the symmetry properties of the system in the
standard sense belongs to the situation characterized by the above
theorem. For, if under a symmetry transformation in the "physical
space" the physical states are unchanged, we obtain automatically
two equivalent descriptions in dv, one corresponding to the
original and the other to the transformed frames and these two
descriptions must be related to each other by unitary {or anti-unitary)
transformations. Conversely, and this is more important from our
point of view, the Hilbert space of states must be isomorphic to the
carrier space oT unitary (or anti-unitary) representations of the
symmetry transformations (they may form a group-or an algebra,
etc.), Note that we wish to obtain a concrete Hilbert space to calculate
transition probabilities. Thus, if we know the symmetry transformations
of the system we can start from an arbitrary collection of irreducible
unitary (or anti-unitary) representation spaces of the symmetryrr,
transformations to build up the Hilbert space %>. This solves the
problem partly, but not completely because we do not know what
collection of irreducible representations we have.
1. 8 Uniqueness of operators
We have said that the vectors from the rays of two equivalent
descriptions can be so chosen that the mapping of vectors if/. <—> \fj^
is either unitary, or anti-unitary. There is one other important
phase problem in quantum theory and this concerns the uniqueness of
the unitary (or anti-unitary) correspondence ^.
- 5 -
1. 9 Ray representations
If there are two equivalent descriptions with rays ¥,,<!>, . . .
and T , $ , . . . , respectively, corresponding to the same states as
seen by the two different descriptions (passive view) or with rays
¥•,*$,, . . . corresponding to states la } in the first description and
to the transformed states {Ts,} in the second description (active view),
then we know that we can choose vectors \p c ¥ , . . . \Jj e Y . . .
such that
^2 = UT ^1 ' ^2 = UT Vl' ' " ' ^
That is, if ip is a vector of Y , then U^ tp is a vector of the ray ¥ .
Now if there are two operators UT and U with the property (1), they
can differ only in a constant factor of modulus 1. This result has an
implication on the group law of transformations. For the product
gives the same result as the transformation U —. Consequently,
(2)
Because U is a representation of the symmetry group, the group
law for the representations is more general than the group law itself
ST = ST. Representations of the type (2) are called "ray representations"
or "representations up to a factor". This is again the result of the
fact that we have a correspondence between physical states and rays
in Hilbert space, not vectors.
1.10 Covering group
The ray representations of groups defined by eq. (2) can be
interpreted as ordinary vector representations of the extended group,
the so-called quantum mechanical covering group. (Appendix II).
Summarizing,we have: A symmetry group Q of the physical
system induces a group Uo of invertit>le mappings of Jy on to itself,
which is unitary or anti-unitary and is a representation of the coveringItgroup of G »&nd Jy is an arbitrary collection of irreducible carrier spaces
of U s .
- 6 -
1.11 Continuity
Moreover, if some sort of topology is defined in (r , telling us
which symmetry transformations are close to each other, then the
postulate of the continuity of the physical transition probabilities
implies that the mapping S «-» U must also satisfy suitable
requirements of continuity. Thus, we have to do with continuous
representations of groups.
1.12 Unitary and anti-unitary operators,
The group property of the transformations, eq. (2),and
continuity allow us to determine the unitary or the anti-unitary
character of U_.
If for every group element S of G we have
S = s2 (3)
where s is also a group transformation, we have
Ug= u(s) U^ . (4)
The square of an operator is unitary, whether the operator is unitary
or not. Thus, all !!„ is unitary. For group i connected continuously
to identity, eq. (3) is satisfied and they will be represented by unitary
operators. For the anti-unitary case, eq. (3) must break down. If
eq. (3) does not hold, further considerations are necessary to decide
the unitary or anti-unitary character of U_ (see Appendix III).
In the unitary case, one can define a normalized operator U^
that U l = U Then U3S-1
transformations.we have from (2)
such that U l = U Then U3S-1 =* w(S,S~ ) l . For two commuting
USUT=C(S.T)UTUS , O(S.T). {£f |[only if Ufp and UR also commute J
and we find C(S, T) = +1 / In general, if the commutator-1 -1 *U_U U U (which is independent of the normalizations of U,- and
1 S I S •*•
U- and which is uniquely determined from U— and U ) is a multiple
_ rt
of I, i. e., C = TST" S"1 = I, then VQ = w(T, S) I, and u(T, S) is a
characteristic of the coherent subspace only, i. e. , has a unique value
in each coherent subspace. If U and IL are members of the same one-
parametric subgroup then(j(T,S) = 1.
1*13 Superselgction rules again
In Seo.1.2 we have seen that the vectors ^ oc^ l n andn
V 7 06 (A. y ) "belong to different rays (states) although ^ and
X y belong to the same ray. If a physical symmetry transformation
of the system changes W into A W then, "because the state of the' n n ' n '
system has not changed, a superposition of the form (_^ <*n y^ isn
not possible, unless A a 1, • The relative phase A between vectors
in different coherent sectors is not observable because the physios
has not changed under the symmetry transformation. No physical
measurement can distinguish the state 2_s a n yn from the staten
ny at (X (f ), Thus, to show the existence of a superselection rulenwe need a symmetry transformation (a physical postulate) and the
existence of vectors V which go into A W under this transformation,' n n ' n
Example 1, - Rotational invariance and fermion superselection rule
Consider,for- concreteness, a state ^, •» | "|»m^ belonging to
tbe representation J>? and a state fL « In ,m^ belonging to the
representation I) ", n » integer* Consider a rotation by the Euler
angles (0,^,0). The states transform according to eq,(A.IV. 52),
i.e.,
W ^ Id,*') .
Now for m 2irt we have
Thus, we get an extra phase of (-1) in the linear combination of our
two states} hence, according to our previous discussion, there is a
superselection rule between the states with integer j and those with
half-odd integer 5"*values,
-8-
The rotation "by 2Tr in this discussion can also be replaced
"by the commutator of two rotations "by IT about axes perpendicular to
each other*-*. These two operators commute and their commutator is
independent of the normalization of these operators. One again
finds, taking DQ = (n.^) D(n2Tr) D ^ n ) "1 D^ir)""1 (n = (n^^i ) . unit
vector), that D^ - (-l)2^. Thus DQ is represented by a phase factor
whose value depends on the coherent subsp'ace (see Seel .12).
Remarkt The postulate of "dynamical groups" which says that the
collection of irreducible representations of symmetry groups is itself
a particular representation of a larger non-compact group (e.g. SL(2,C))
automatically incorporates this superselection rule, because in the
representations of these higher groups, either only integer, or only
half-integer representations of SU(2) subgroup occur, (see Ch.II).
Example 2 - SU(2) group for isospin and superselection rules
If the STJ(2) group describing the isotopic spin raultiplets of
particles were an exact symmetry group of nature as the SU(2) group
for spin, then by the result of Ex.1 there would be a superselection
rule between the integer and half-odd integer I-spin states. Now for
strong interactions which are independent of the electric charge, SU(2)^
is a good symmetry group. This means that there are no 2ure_ states
of the form | Y. / + j A \ , for example. There are,however,
pure states like j m.y + | p ^ • These superpositions violate the
superselection rule on charge (see Ex,3 below); consequently there is
no superselection rule for charge for strong interactions alone. In
the presence of electromagnetic and weak interactions, SU(2)_ is not
a symmetry Croup, but then charge superselection rule holds, a pure
state I XT / + A ^ now exists, but not a state
Y- y + I A* \ . In fact, an 311(2 rotation taking Z +
into /i (or n into p) does not leave the system unchanged but
corresponds to the weak interaction process: Z -> A TT (or
n -> p + e + V )•
Similarly, if a hypothetical "superweak interaction" violates
the rotational invariance, then we can have pure states of the form
|N^> + I TT > , and the reaction N — » TTA, where A are the quanta
of this new interaction.
G.C. Hegerfeldt, K. Kraus and E.P, Wigner, J. Math. Phys.j), 2029 (1968).
-9-
Example 3 - Superselection rules for gauge groups
Two equivalent descriptions obtained from each other by a
commutative one-parameter continuous group (not obviously related to
space-time transformations) implies the existence of an additive
quantum number q, and the eigenstates transform as
For two-states with different values of q, e»g»> +1 and ~1» we
obtain two different phases e and e , hence a superselection
rule for q. The basic physical assumption underlying all such
selection rules, such as electric charge, baryon number, lepton number,
we repeat, is the requirement that the multiplication of all states
by e produces no observable change in the system, hence equivalent
descriptions and gauge groups.
One can form, instead of pure states, mixed states out of vectors*)
from different coherent subspaces. But this will not interest us here <
1,14 Implications of the Buperselection rules on parity and othergroup extensions
Within a coherent subspace the parity of each state (relative
to one of them) is well determined. In fact we use the ray
representations of the full 0(3) group, or the full Lorentz group,
including reflections. In this case the parity is defined either in
the same representation space as SO(3) (or proper homogeneous
Lorentz group) or in a doubled Hilbert space (see Apps. II & IV ) .
Thus relative parities are well determined, e.g. for the levels of
H-atom, and for particle-anti-particle pair in Dirac theory. However,
for states in different coherent subspaces, the relative parity is
not determined because we oannot take a linear combination of two
such states and see how it transforms under parity. Note that the
measured relative parity between K and Tr is actually that between it
and the deuteron (*J » 1 ) ,
f Formal structure of quantum theory with superselection.rules, seet
J.M. Jauch, Helv. Phys. Act a ^ , 711 (i960) and J#M. Jauch and
B. Misra, ibid, 2£, 699 (1961).
-10-
Very similar considerations apply to other group extensions,e»g«» ^7 charge conjugation.
The extension of the isotopic spin gr^up SU(2) by a reflection
operator implies a doubling of I a n/2 states, but not necessarily
of I s nf n a integer states. Now this extension is carried out by
C a charge conjugation, or by G s C e 2, [The use of G and G,
respectively, corresponds in the rotation group for spin to the use
of TL and parity P$ G commutes with all isospin rotations as P
commutes with all space rotations (App. IV. 7) r . G tells us
whether we have polar or axial vectors in I-spaeej e.g., f is a
polar vector. Therefore the doubling with G takes us to antiparticles.
Consequently we have among others the result that I m n/2 boson—
multiplets cannot contain antiparticles; they must lie in the other
half of the doubled space *\ In the limit of an exact STJ(2).j. the
relative G-parity (isospin-parity) between I = n/2 and I = n multiplets
is not defined} nor is it defined between states with different
charges or baryon numbers. It is defined, however, between, e.g.,
I . 1 multiplet (ir) and two I = •g-multiplets with N = O(e.g.
1.15 "Irreducibility postulate" for symmetry groups.
We said in Sec.1*7 "that the concrete Hilbert space (CHS) for
our system is a collection of irreducible unitary (or anti-unitary)
representation spaces of the symmetry transformations. Can this
information be of any use before we have determined the complete CHS?
*) This result, proved here group theoretically, has been the subject of
many recent papers where it was proved either from the assumption of
local field theories (P* Carruthers, Phys. Rev, Letters l£, 353 (1967)),
or from analyticity (crossing), (H, Lee, Phys. Rev. Letters 18, 109-8
(1967)). A proof similar to ours but using OPT was given, B. 2umino
and D. Zwanziger, Phys. Rev. I64, 1959 (1967).
-11-
The answer is yes, if we know the properties of a physical quantity
as a particular tensor operator (see definition in Appendix IV) with
respect to the symmetry group. In that case, partial information
relating to the dependence of the matrix elements on the states can
be derived from the commutation relations of tensor operators
(eq. A. IV. 45 ), In ordinary quantum mechanics the concept of
symmetry is used in the narrow sense to mean the symmetry of the
Hamiltonian. The corresponding group is the group of degeneracy
of the energy. One then introduces an additional physical postulate,
the socalled "irreducibility postulate" which says that each eigen-
space of the energy is an irreducible carrier space of the maximal
symmetry group. Only then can one relate properties of states
within each multiplet of G. We can see that the "dynamical group"
approach explains this "postulate" rather naturally (see the remark
in Sec. 1.13 after Ex. 1).
-12-
2. QUANTUM DYNAMICS
In Sec. 1. 5 T have defined "quantum dynamics11 to be the
determination of the concrete Hilbert space (CHS) of a physical
system in order to evaluate the probability amplitudes as the scalar
product of two state vectors. In Sec.l. 7 it was shown that this CHS
must be a collection of carrier spaces of the unitary (or anti-unitary)
irreducible representations of the symmetry groups of the system.
To proceed further it is important to distinguish between an
isolated system and a system plus interaction. An isolated system
has a CHS which is a single irreducible representation of the
geometrical symmetry groups (such as the space-time symmetry
groups). External interactions cause transitions to other states;
they cause the system to reveal its internal structure. We must then
consider the system together with interactions (i.e., with other
systems) as a larger unit which now possesses in terms of the new
relative co-ordinates a larger symmetry than the original system.
According to this plan I shall begin with the simple quantum
systems and simple interactions and then define a new generalized
concept of symmetry, the dynamical symmetry, and then give a group
theoretical formulation of quantum dynamics. Eventually the boundary
between "kinematics" and "dynamics" may completely disappear.
2.1 Definition of simple quantum systems
Using the equivalence of descriptions provided by the symmetry
transformations we can already define special quantum systems by
the irreducible representations of the symmetry groups. Such simple
systems are approximations to the more realistic physical sjstems,
under certain conditions, when the external interactions restrict the
complete freedom of the system (or when those simple interactions
are involved which only cause transitions to a limited number of
states out of all the possible states).
-13-
Consider the rotational invariance and the concrete Hilbert
space (CHS) resulting from the irreducible representations of the
rotation group.
Definition: A j - rotator is a system whose states belong to the
irreducible D -representation of the rotation group. The Hilbert
space is (2j + l)-dimensional and a basis is labelled by jj;m)> ,
j = fixed [Appendix IV] .
A j - rota tor cannot move (rest frame) and does not have any
other degrees of freedom . The system is completely characterized
by the statement that any two descriptions related to each other by
a rotation a re equivalent (Sec. 1.6). Indeed, if <p = \ qj | m)>- ,
Z Z_j r n
ip ] m y a re arbitrary states, the description of the system
by the states |(p, ip, ... I and by the states j DJqj, D ip, . . . 1 are
equivalent:
(Dj<p,D3<&) = (qj.D
m
The following considerations apply equally to an I-isorotator, a
system with isospin I whose other quantum numbers do not enter
into the processes considered.
The system can be easily generalized to include discrete symmetries.
For example, if we include reflection symmetry as well, that is,
consider the full group O(3), then the parity quantum number is
introduced. The parity of the state Jj;m)> can be assigned as
(-1) , for j = integer and the Hilbert space need not be doubled; for
3 = half-odd integer, we have to double the Hilbert space in order to
define parity eigenstates (Appendix IV.7. 3).
-14-
• • v , >*• • •
External interactions allow us to observe the change of the
state in such a system. When these are small enough not to form
new systems (i.e., interactions which do not "break" the rotator ),
they can be represented by operators acting on the carrier space of
D . The simplest interaction is one which just rotates the rotator,
in which case this active point of view is equivalent to a passive
point of view of a rotation of the co-ordinate frame. The effect of
an external interaction will be represented by the "interaction
vertex"
Interaction J
and the matrix element
M = < m 1 ) . j | m > - (2.1)
is the transition probability amplitude that the measurement J (i. e.
interaction) will cause a transition from state m to the state m1 .
This is the only measurable quantity; the wave function itself need
not be introduced. If, for example, the measurement is a
rotation and m is the value of the angular momentum along the z-axis
of the system before the measurement, then
< m ' | e ^ " ? ] m > = D ^ M ) (2.2)
is the probability amplitude that after the rotation the angular
momentum has the projection m' . Clearly, the sum of
probabilities over all final states is \ JD J | = 1 . Accordingr i_~i m m
m1
to the fomulas (A. IV. 52), we can introduce rotated states
(with respect to a fixed direction) ] m j 0 > = e * | m > , then (2. 2)
is the overlap ( m 1 , 0jm> = <m ' |m , -
When the interaction "breaks" the rotator, the Hilbert space must be enlarged to include the
degrees of freedom of the constituents.
- 1 5 -
More generally, the interaction vertex may be described by a
tensor operator (A. IV. 11) so that
M - (2.3)
for example, a "vector vertex" may be of the form
M - g
where g is a coupling constant and the matrix elements <( m" j j j m
have been evaluated in {A, IV. 54).
Note that we have not introduced the notion of the "time-
evolution of the state", nor the Hamiltonian. The states of the
system are asymptotic states labelled by jm^ only (j is fixed).
An experiment is an interaction vertex and we evaluate the
probability of transition to the final state. This formulation is the
same for relativistic and non-relativistic systems. The example
of the j-rotator is indeed very simple and restrictive. More
realistic physical systems undergo transitions with a change of
mass (or energy), of momentum and of other properties under the
interactions. In order to account for these properties we have to
enlarge the Hilbert space and the symmetry groups. We shall
consider systems with increasing complexity as shown in the
following Table:
Rotation group
Galilei group
Poincare' group
(0(4)Higher groups JO(3,1)
l0(4,l)
Higher groups $ndPoincare group
System endowed with one spin only
Non-relativistic system endowed with spin, energy andmomentum
Relativistic system endowed with spin and four-momentum
System at rest with many spin states and other degrees offreedom
Relativistic systems with many mass and spin states
-16-
2. 2 Elementary systems
We now go over immediately to the largest symmetry groups
associated with the geometrical transformations of space-time: the
Galilei group or the Poincare* group. All the transformations of
these groups have a physical, geometrical interpretation:
a) space and time translations (displacements of the co-
ordinate frame);
b) rotations and reflections;
c) transformations giving a system a velocity ("boost"
transformations).
An isolated system must allow equivalent descriptions under
the Poincare group. Consequently we can define elementary systems
whose concrete Hilbert space (CHS) is the car r ie r space of a single
irreducible representation of the full Poincare' group $>. An
elementary system is characterized by the invariants of $, mass (m )
and spin (j( j"+l)) or helicity. The eigenstates of the displacements
are labelled by |p ;or|> ; the rest frame states Ip^p*55 O;cr)> ,
= O) a re rotational invariant and a velocity imparting is given by
if • M
Here M are the generators of pure Lorentz transformations for the
rest states and § = pen — = psh — for massive particles.
m m
An elementary system may reveal under external probing a
more complex internal structure. We can give an operational
definition of an elementary particle . To do this we first say that
two elementary systems are connected if physical interactions can' connect the Hilbert spaces 7v(m , j .) and f ( m j ) of the two_ ^
A.O. Barut, Dynamical groups and a criterion of elementarity, in
Lectures in Theoretical Physics, Vol. K B , p. 273 (Gordon and Breach,
NY, 1967).
-17-
systems. For example, the connection of Is and 2p states of the
H-atom by photo-absorption or the connection of the neutron and
proton states by /3-decay. We have then:
Definition: An elementary particle (EP) is an elementary system
whose states in no way can be physically connected to the states of
other systems. Its Hilbert space is isolated, i.e. , the states of
one elementary particle |l)> do not form a linear space with those
of other systems ] 2 )> , that is,the superposition |l)> + [2 > is
physically meaningless. The only effect that outside interactions
can have on an EP is to change the state within the irreducible
representation, i. e., to change its momentum. It follows that an EP
can have only those internal quantum numbers for which there are
absolute superselection rules (see (A. I. 3) and (A.. 1.13)).
This operational definition of an elementary particle reflects
the dependence of the concept on the nature of interactions, as
it should be. Clearly, in the kinetic theory of gases, for example,
the molecules are elementary particles for, under the processes
considered,the internal structure of the molecule is not excited
and there is no connection to other parts of the Hilbert space.
Similarly, nuclei are elementary particles in atomic phenomena, and
so on.
As we did in the previous section, a "measurement" on an
elementary particle will be described by an interaction vertex
M = <FjOr'jJ]p;cr> - <>'|e J e *
Here m and j are fixed on both sides (hence the label § inside the
state vectors is not necessary in this case). Again J can be a scalar
or a general tensor operator.
-18-
i-t
2. 3 The generalized concept of symmetry
There are some transformations which are important in
determining the concrete Hilbert space of the system, but which
obviously do not correspond to the geometrical transformations
discussed in the previous section. As prototypes/the O(4) group for
the bound states and the O(3,1) group for the scattering states of the
H-atom for a given energy, the SU(3) group for the multiplets of
hadrons and the O(4,1) group for all levels of the H-atom. At first
there seems to be no physical interpretation of the corresponding
transformations. Furthermore, they violate, for isolated systems,
the conservation laws and the supers election rules: we cannot
perform a superposition of a proton and a neutron state, for example.
However, these groups clearly account for the internal degrees of
freedom and for the complete concrete Hilbert space of the
system, in the same way as the Poincare' group accounts for the
external quantum numbers (momentum and spin) of the system. How
are we going then to interpret these larger groups physically?
The clue to this problem lies in the interactions. We see from our
discussion leading to this point that the new transformations referred
to above are not accessible to isolated systems and they make no
sense. But thej are physically meaningful for the larger system
consisting of (system + interaction). We assert now that they
are "symmetry" transformatioreof this larger system.
First of all the clash with the superselection rules and
conservation laws disappears once we identify the new transformations
with the external interactions changing physically the state of the
system. For example, a neutron is indeed transformed into a
proton under the weak forces . For our original system alone
these transformations constitute physical changes, not symmetries;
:f. the discussion in A. O. Barut, Phys. Rev. 156, 1538 (1967).
-19-
they lead to an inequivalent description. But for the combined system
they can be considered as symmetries, in the sense that one obtains
from one possible state of the system another possible state of the
system, corresponding,in general, even to a different total mass and
spin. It follows from our definition of elementary particles in the
previous section that the systems admitting these more general
transformations cannot be elementary. This is also intuitively clear
from the existence of new quantum numbers corresponding to internal
degrees of freedom.
2. 4 Physical interpretation of dynamical transformations
Once the composite structure of the system is taken into account,
it will be possible to give a physical interpretation of the new
transformations. They correspond to active changes in the internal
constitution, changes in the distribution of matter by outside agents.
The characteristic of geometrical transformations was that the active
point of view (i.e., rotating a system) was equivalent to the passive
point of view (i. e., rotation of the co-ordinate frame). This is now no
longer true; there is no simple passive transformation of the observer
which would have the same effect as the external interaction on the
system (see however. Sec. 2. 5).
As an example we consider the O(4, 2) group and the H-atom.
Now the geometrical symmetry transformations connect states of the
atom with the same internal shape, but at different points in space,
at rest or moving with some velocity and with different orientations of
the plane of the orbit. In addition we can now make dynamical
transformations by changing the relative distance between the electron
and proton (e.g. dilatations), by giving an extra tilt to the orbit or
by giving extra energy to the electron (photo-absorption) so that
it moves faster. The important point is that although a continuous,
infinity of such transformations in the internal structure could be made,
all possible states may be obtained from a denumerable basis set .
The same situation occurs, by the way,also for rotations: although
there are infinitely many orientations of the plane of the orbit, there
- 2 0 -
are only 2(2j + 1) distinct basis states out of which all other states
can be constructed. In fact the dynamical group idea suggests
that, in the rest frame of the system, the dynamical transformations
on the system form a relatively simple group.
2. 5 An analogy. General theory of relativity
Although a non-quantum theory, the general relativity offers,
at least conceptually, a complete analogy to our dynamical
transformations. Here we have,in addition to the Poincare group,
physical changes corresponding to gravitational interactions of the
system . Under these interactions the system makes transitions to
new states not accessible to free systems, just like the dynamical
transformations in the H-atom, or the SU(3) transformations changing
neutron into proton. The further important feature in general
relativity is that the active point of view (gravitation) can again be
made equivalent to a passive point of view by enlarging the
transformations of the Poincare'group by all co-ordinate transformations.
Thus changes under gravitation become "symmetry" transformations
on the same footing as changes under rotations or displacements.
The concept of symmetry is now generalized. Equivalent descriptions
of the system now also include descriptions which differ from each
other by the presence or absence of gravitation. In other words
gravitational forces may be eliminated in favour of the dynamical
geometry .
*>
That the symmetnesof general relativity are of dynamical nature is
also the conclusion of Cartan, Fock and Wigner, see
K.M. F. Houtappel, H. van Dam and E. P. Wigner, Rev. Mod. Phys.
37, 595 (1965) for a very detailed discussion of geometrical symmetries
and for other references. See also A.O. Barut and A. B6*hm, Phys.
Rev. 139. B1107 (1965).
-21-
2. 6 Complete and approximate geometrization of dynamics
If the interaction can be completely eliminated as in general
relativity the system and its gravitational interactions are described
completely, in quantum theory, by a single unitary irreducible
representation of the generalized symmetry group (dynamical group).
The transition probability under gravitation is just the overlap
between the two relevant states )O and ]2)>
M = <1|J]2> ;
in particular we expect J = 1.
If, however, we have not eliminated the complete interaction in
the Hamiltonian
H - H0 + H l + H 2 '
but a part H + H only and succeeded in obtaining the states under
H + H from a dynamical group, then the interaction operator in
M '
will be quite simple in contrastto the situation when only H~ would
have been replaced by symmetry transformations. This is indeed
the case. In the example of the electromagnetic interactions of the
H-atom, for example, the use of the larger dynamical group O(4, 2)
brings forth that the interaction operator is to a large extent a
constant operator, V , with a small part depending linearly on the
momentum.
- 2 2 -
2. 7 The concept of "interaction" versus global point of view inrelativistic theories
The dominant point of view in microscopic theories is based
on the separation
system + system + interaction
This view presupposes that the systems under consideration still
preserve somewhat their individuality even under the interaction.
Whether such a separation is possible for relativistic systems is
not clear. There are some well known difficulties connected with
the proper times of the systems even in classical relativistic
mechanics. Besides these fundamental difficulties, a theory based
on such a separation, in the relativistic case, always leads to an
infinite system of equations; we know this from our experience in
perturbation theory or in S-matrix theory. In axiomatic field
theory J, or in Heisenberg's non-linear theory the above
separation is not made, but one operates with interpolating or with a few
self-coupled basic fields. The dynamical group approach, on the
other hand, deals with the observable quantum numbers only; it is a
global description of interacting systems in which the concept of
interaction (e. g. potential) has been eliminated in favour of total
observable quantum numbers and their range. There is thus the
possibility that the behaviour of the system can be describedthe
relativistically in finitely many (few) steps. The example of [relativistic
H-atom and positronium in which the recoil effects are evaluated
globally is very illustrative % Thus the approach of dynamical
groups is not only an algebraic reformulation of the present theories,
but may also help to solve some of the difficulties of the relativistic
quantum theories. •!•_
H. Lehmann, K. Symanzik and W. Zimmermann, Nuovo CimentoU 425(1955),-A.S. Whightman, Phys. Rev. 101, 860 (1956).
* *'W. Heisenberg, Introduction to the Unified Field Theory of Elementary Particles (London, Interscience,1966). i.
* * * ) A . O . Barutand A. Baiquni, ICTP, Trieste, preprint IC/69/3.
- 23 -
2.8. Survey of further chapters
The ideas expressed in this chapter will be extended to larger
and larger groups. According to the Table in 2.1, we shall considerand
the concrete Hilbert spaces of the Poincare' groupithose of the larger
groups containing the internal quantum numbers, together with the
corresponding physical systems in which dynamics has been
expressed as "geometry".
-24-
APPENDIX I
WIGNER'S THEOREM ^
Let the rays tjr , $.. , . . . represent the states of the system in
one description and "(jr , $ 0 , . . . represent the same states in the second
description.
Let ^ , (p1 , . . . and 0 , <p , . . . be two sets of orthonormal
bases chosen from the first and second set of rays. We are given a
transformation which preserves the absolute values. The problem is
to construct a corresponding transformation U on the vectors by the
suitable choice of the phases. It could be a priori that U0.. = c^09 ,
Uqpn = c <p , . . . and that no relations between the c's can be es-1 <p 2
tablished, in which case U is not even a linear operator. We want,
in fact, to show 1hut U can be so defined that it is a unitary or an anti-
unitary operator.
We single out the unit vector 01 and choose U^i = 0O . This& 1 1 2
is the only arbi trary choice and shall show that all other phases are
uniquely determined. Thus U is determined up to an overall phase
factor.
Next consider the vector 01 + <p1 . A representative vector
of the corresponding ray in the second description is aip + b<p .
We have then
b'cp2
where we must have c = I /a by the previous choice, and put
cb = b1 = b/a . We now define Uep. by 11(0. + qpj - 0 , or simply
*> E. P. Wigner, "Group Theory",(Academic P re s s , N. Y., 1959) Appx. 20.
For more details see V. Barg'mann, J. Math. Phys. J5, 862 (1964);
U. Uhlhorn, Arch. Fysik, 23, 307 (1963).
-25-
by b'<P9 • Hence
<p,) = Uf + U<p.
Similarly for a general ^ = a ^ + a ^ + , . . we choose a represent-
we getRk^ = cos0 6k^ + (1 -cos0) n V - sin0 £k^ j nj . (A. IV. 5)
One can verify that RR = I or R R = 5 ; using the identities
t t j l t H
Thus R is orthogonal (unitary and real), det R = +1 , i. e., R € SO(3),
and is identical with the unitary irreducible representation D (n*0).
The group law is
R(n\0) = R^ejR^ej ,
fl 1 9 1 9
cos- = cos-y cos~^" " " i ' " 2 Sin~2~ Sin~2~
(A. IV. 6)
ii sin~ = ft sin— cos— + n sin-r- cos— + n,x fi sin— sin—£t \. ct Ct ct ct it £t Ct £t
This result can be put in a more elegant form. Let
-32-
, ?= f i s in | , 0 ^ 0 4pr (A.IV. 7)
2 -*2 JH _>2 V /^^-^
with a + a =1 , a = + l l-a ,xe., the group space isl/the 4-dimensional
unit sphere/ The group of motions of the group space S is O(4). The
law of imposit ion (A. IV. 6) now reads
3 =«<2
If we now associate a real quaternion to the three group parameters by
o <y oi - j = k = - 1 , i j = - j i = k (cyc l ic ) ; aQ, a^ j
(A. IV. 8)
then the group law becomes
Thus we have obtained a representation of SO(3) in terms of quaternions:
q = 1 is the identity rotation, q = a - ia - jot - ka is the inverse
rotation; qq = *qq = 1 .
*") For complex a , a. we can get a representation of the Lorentzp
group.
-33-
3, Passage to the 2-dimensional double-valued representation
The quaternions can be represented by 2 x 2 matrices via the
correspondence
io J-H>-HT k^>-i<T '
q-> aQ - i ^ s + U . (A. IV. 9)
We have then U+U = UU+ = (a - ia-c)(a + ia-3) = a2 + B2 = 1 .
Thus we have a representation of the rotations in terms of the matrices
of SU(2)O In terms ol the original parameters {A. IV. 7):
+U = an - !<?•? = cosf - io*-n*sinf = e" 1 ^" °l2
This can also be written as
n !<?? cosf ionsinf e . (A. IV. 10)
±U = (^ h\ , | a | 2 + | b | 2 = l (A. IV. l la )
where the new parameters of the rotation group a re
9 . . 6 , . . v . 9a = cos- - i n s in- , b - (- i n - n ) s in- . (A. IV. l ib)
ii O Z X £, £i
Another set of parameters, i. e.,the real Euler angles, characterizing
the rotations is given by
a = e i (* + ^ 2 cosf , b=e*~v)!2si4 . (A. IV. 12a)
2 + -1<r. = 1 , t r c = 0 , tr. = or. = or. , <r x a = 2i a or
X 1 1 1 1 ~ ~ —
CT.CT. = 5 . . + i £. or,l j 13 ljk k
( a * c r ) ( b * < 7 ) = ( a « b ) + i ( a x b ) ' t r
Z ( a , ) . (or, ) . = 2 6 , 6 U - 6 ^ 6 ,k a b k c d a d b e a b c d
k
t r ( o \ o \ ) = 2 6 . . ; t r { < r . c r . o \ ) = 2 i 6 . . ' t r ( o - . c r . c r . c r ) = 2 ( 6 , . 6 t - 6 . , 6. + 8 .i ] 13 1 ki jkZ ] k i m lk ^m it km 3m